284 New Algebraic Series. 
The series of the first member, as is well known, is in- 
commensurable (which is easily proved, amoug other meth- 
ods, by the theory of continued fractions) and comprised 
between 2 and 3. Itis the base of Napier’s System of Log- 
arithms. In representing it by e, as usual, we shail have 
Ax ct  Atx?. \ Ate 
Se ea es 78 
3 
Ks &e. 
Ae ey 
Ifwe make ¢ =a, in which case A will be the Jogarithm 
of a, according to Napier’s System, (or la,) we shall have 
one _ 
FE as sd o@t 1808 te 
_A formula which gives the developement of exponen- 
tials iato series or of any number a, in a function of its lo- 
garithm. If in this latter formula » be changed into m, and 
a, into 1+, it will becom 
y ml{1+-2)  m?l?(1+2) m3l3(1+2 Rie. 
(ee seh a ge 3 Ee CEA 
But it has been already shewn that 
‘ 2 
(1+2)"=1+m;+m(m—=1) = 5+ &c. whence 
mi2(1-+2) m?13(14 =. 
12) + : 42°01 the. =j+(m—1) ty. 2 
3 
+(m—1) (m — 2) Tit &e. 
By making m=», in this last equation, we have 
Ki baler feet ke 
Lee ola BS ter tie 
A formula which gives the hyperbolic or naperian loga- 
i of 1+, in a function of the number x: 
A variety of other important applications of the general 
principle, will easily suggest themselves, particularly 19 the 
developement of circular functions into series, &c- 4A 
branch of analysis of the greatest importance in moder As- 
nomy and Physies. 
